Help with Beginner Index Notation

[fttteotd]

Okay, so I'm learning some basic index notation, and I have a few questions...

Homework Statement

f= scalar field
F = vector field

so, we are supposed to show that curl(fF) = fcurl(F) + ($$\nabla$$f) x F

The Attempt at a Solution

curl(fF) = [$$\nabla$$ x (fF))]$$_{k}$$ = $$\in$$$$_{ijk}$$(f$$\partial$$$$_{i}$$F$$_{j}$$ + F$$_{j}$$$$\partial$$$$_{i}$$f)

any help??

(all the superscripts are supposed to be subscripts, i dunno)

 

[tiny-tim]

Welcome to PF!

Hi fttteotd! Welcome to PF!

(have a curly d: ∂ and a nabla: ∇ and an epsilon: ε and use itex rather than tex in the middle of a line )

[∇ x (fF)]_i = ε_ijk∂_j(fF_k) …

and now use the chain rule!

 

[kerimek]

Sure, I'd be happy to help with your beginner index notation questions. It looks like you're on the right track with your attempt at a solution. Let me break down the steps for you:

1. First, let's define the curl of a vector field F as the vector field given by the cross product of the gradient operator (\nabla) and F. In index notation, this can be written as [\nabla x F]_{k} = \in_{ijk}\partial_{i}F_{j}.

2. Next, we can use the product rule for differentiation to expand the expression for curl(fF). This gives us [\nabla x (fF))]_{k} = f[\nabla x F]_{k} + F_{j}[\nabla x f]_{k}.

3. Now, we can substitute our definition of the curl of a vector field from step 1 into the first term of the expanded expression. This gives us [\nabla x (fF))]_{k} = f(\in_{ijk}\partial_{i}F_{j}) + F_{j}[\nabla x f]_{k}.

4. We can then use the product rule again to expand the second term, [\nabla x f]_{k}, which gives us [\nabla x (fF))]_{k} = f(\in_{ijk}\partial_{i}F_{j}) + F_{j}(\in_{ijk}\partial_{i}f).

5. Now, we can rearrange the terms in the second term to get [\nabla x (fF))]_{k} = f(\in_{ijk}\partial_{i}F_{j}) + (\in_{ijk}F_{j}\partial_{i}f).

6. Finally, we can use the definition of the cross product to rewrite the first term, \in_{ijk}\partial_{i}F_{j}, as the curl of the vector field F. This gives us [\nabla x (fF))]_{k} = f(\in_{ijk}\partial_{i}F_{j}) + (\nabla x F)_{k}\cdot f.

7. And there we have it, we have shown that curl(fF) = fcurl(F) + (\nabla x F) x f, as required.